Polynomial & Rational Inequalities
A polynomial inequality is any inequality of the form p(x) > 0, p(x) < 0, p(x) ≥ 0, or p(x) ≤ 0, where p(x) is a polynomial of degree two or higher. A rational inequality involves a ratio of two polynomials, such as (x² − 1)/(x + 3) < 0. Unlike simple linear inequalities, these problems require a systematic approach because the sign of the expression can flip multiple times across the number line. The sign chart (also called a sign analysis or sign diagram) is the central tool for solving these inequalities. The key idea is that a polynomial or rational expression can only change sign at a zero of the numerator (where the expression equals zero) or at a zero of the denominator (where the expression is undefined). Between any two consecutive critical numbers, the expression maintains a constant sign. By testing one value in each interval, you determine the sign everywhere. Mastering polynomial and rational inequalities builds directly on factoring, understanding multiplicity of zeros, and careful attention to domain restrictions. These techniques appear throughout precalculus, calculus (where sign charts determine where a derivative is positive or negative), and real-world modeling problems involving areas, distances, and rates.
- Move Everything to One Side
- Find All Critical Numbers
- Consider Multiplicity
- Build the Sign Chart
- Write the Solution Set
Move Everything to One Side
Rewrite the inequality so that one side is 0. For example, x² + x − 6 > 0 is already in this form. For a rational inequality such as (x+2)/(x−1) ≥ 2, subtract 2 from both sides first: (x+2)/(x−1) − 2 ≥ 0, then combine into a single fraction: (x+2 − 2(x−1))/(x−1) = (−x+4)/(x−1) ≥ 0. Never cross-multiply a rational inequality without first knowing the sign of the denominator.
Find All Critical Numbers
For a polynomial, factor completely and set each factor equal to zero. For a rational expression, set the numerator equal to zero (giving candidate solutions) and set the denominator equal to zero (giving excluded values where the expression is undefined). List all critical numbers on a number line. These points divide the line into intervals.
Consider Multiplicity
The multiplicity of a zero determines whether the sign changes at that point or stays the same. If a factor (x − c) appears an odd number of times, the expression changes sign at x = c. If it appears an even number of times (e.g., (x−c)²), the expression touches zero but does not change sign — it stays positive (or non-negative) on both sides. This lets you fill in a sign chart without test values in some cases.
Build the Sign Chart
Draw a number line and mark all critical numbers. Choose one test value inside each interval and evaluate the full expression (or just track the sign of each factor separately — faster for factored forms). Record a + or − in each interval. For a rational expression, also record the sign at denominator zeros as 'undefined' (vertical asymptote).
Write the Solution Set
Select the intervals where the sign matches the inequality (e.g., positive intervals for > 0). Then decide whether to include the endpoints: include a critical number if it is a zero of the numerator AND the inequality is non-strict (≥ or ≤). Exclude it if the inequality is strict (> or <) or if it makes the denominator zero. Express the answer in interval notation or set notation.
The Key Equation
- p(x) — Numerator polynomial (dimensionless): A polynomial whose zeros are the x-values where the rational expression equals zero; these are candidate solutions.
- q(x) — Denominator polynomial (dimensionless): A polynomial whose zeros are excluded from the domain; they create vertical asymptotes or holes and are never part of the solution.
- \gtrless — Inequality symbol (—): Represents any of the four inequality relations: >, <, ≥, or ≤.
Worked Example
Problem. Solve the rational inequality: (x² − 4) / (x + 3) < 0
Answer. The solution set consists of all x in (−∞, −3) union (−2, 2).
Origins · 1 minRené Descartes and how this was discovered
Descartes developed the Cartesian coordinate system and laid the algebraic foundations for polynomial theory. His 'Rule of Signs' (published in La Géométrie, 1637) gives an upper bound on the number of positive and negative real roots of a polynomial by counting sign changes in its coefficients — a direct ancestor of the sign chart technique used today. Descartes spent most of his adult life in the Netherlands, preferring to sleep until 11 a.m. daily — a habit he credited for his clearest philosophical thinking. He only gave this up when Queen Christina of Sweden summoned him to Stockholm and insisted on 5 a.m. philosophy lessons; he died of pneumonia shortly after.
Related Concepts
- Solving Linear Inequalities — Linear inequalities introduce the core rule that multiplying or dividing by a negative number flips the inequality sign — the same rule that makes sign analysis necessary for higher-degree problems.
- Factoring Polynomials — Sign charts depend entirely on writing the polynomial in fully factored form so that each critical number can be identified and the sign of each factor tracked separately.
- Rational Functions and Asymptotes — Understanding vertical and horizontal asymptotes of rational functions clarifies why denominator zeros are excluded from solutions and how the function behaves near those points.
- Zeros and Multiplicity of Polynomials — Multiplicity determines whether the sign changes or stays the same at a critical number, allowing you to complete a sign chart efficiently without testing every interval.